Bessel Type Function $J^{\theta}_{\beta}$, Bessel Type Operator $\Delta^{\theta}_{\beta}$ and Fractional Fourier-Bessel Type Transform $\mathcal{F}^\theta_{\beta}$

Abstract

Fractional Fourier-Bessel type transformation is defined. Then using these transformations the pseudo-differential Bessel type operators $\mathscr{B}^{\theta}_{\beta,~a}$ is also defined. After that we introduce some class of symbols, Sobolev and Bessel type potentials spaces. Properties of these transformations and operators are investigated.